ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53, 34, 70 Matveev, Vladimir S., Max-Planck-Institut f. Mathematik, Bonn, Germany (temporary), Chelyabinsk University, Russia (permanent) Topalov, Peter Jordanov, Institute of Mathematics and Informatics, BAN, Sofia, Bulgaria Geodesic equivalence and integrability Let G = (g_{ij}) and Ĝ = (ĝ_{ij}) be two smooth metrics on the same manifold M^{n}. They are geodesically equivalent if they have the same geodesics (considered as unparameterized curves). We show that the existence of a (non-trivially) geodesically equivalent metric Ĝ allows us to construct canonically n commuting quadratic in velocities integrals of the geodesic flow of the metric G. Corollary. Let M^{n} be a closed real-analytic manifold supplied with two real-analytic metrics G, Ĝ such that the metrics G, Ĝ are geodesicaly equivalent and strictly non-proportional almost everywhere. Then the fundamental group π(M^{n}) of the manifold M^{n} contains an Abelian subgroup of finite index.